{
  "id": "friction",
  "version": "2ef8f021f803",
  "status": "published",
  "name": "Friction Calculator",
  "question": "What is the friction force?",
  "summary": "Friction calculator: the friction force f = μN, static or kinetic, from a coefficient and the normal force (or a mass on level ground or a slope), or the coefficient μ = f ÷ N.",
  "category": "physics",
  "subcategory": "mechanics",
  "url": "https://www.acalculator.org/physics/friction-calculator",
  "markdown": "https://www.acalculator.org/physics/friction-calculator.md",
  "kind": "function",
  "method": "f = μ × N (kinetic; static friction up to μ_s × N); N = m × g, or m × g × cos θ on a slope; μ = f ÷ N; on a slope it holds while μ_s ≥ tan θ, and slides with a = g(sin θ − μ_k cos θ).",
  "assumptions": [
    "Friction is μ times the normal force and does not depend on the contact area or (for kinetic friction) the speed; the coefficients of Table 6.1 are approximate.",
    "On a slope, the object rests on the plane with no other forces pressing it in or pulling it off."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "find": {
        "title": "Find",
        "description": "Work out the friction force, or the coefficient of friction from a measured force.",
        "type": "string",
        "enum": [
          "force",
          "coefficient"
        ]
      },
      "kind": {
        "title": "Friction",
        "description": "Kinetic friction acts on a sliding object; static friction holds a still one, up to μ_s N.",
        "type": "string",
        "enum": [
          "kinetic",
          "static"
        ]
      },
      "surf": {
        "title": "Surfaces",
        "description": "The two surfaces in contact, which set μ (OpenStax Table 6.1).",
        "type": "string",
        "enum": [
          "rubberDry",
          "woodWood",
          "waxedSnow",
          "metalWood",
          "steelDry",
          "steelOiled",
          "teflon",
          "bone",
          "shoesWood",
          "shoesIce",
          "iceIce",
          "steelIce",
          "custom"
        ]
      },
      "mu": {
        "title": "Coefficient of friction (μ)",
        "description": "The coefficient for the two surfaces, a pure number.",
        "type": "number",
        "minimum": 0,
        "maximum": 10
      },
      "f": {
        "title": "Friction force (f)",
        "description": "The measured friction force on the sliding object, or the largest static force just before it slips.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "force",
        "minimum": 0,
        "maximum": 1000000000000000
      },
      "by": {
        "title": "Normal force from",
        "description": "Type the normal force, or work it out from a mass on level ground or on a slope.",
        "type": "string",
        "enum": [
          "mass",
          "incline",
          "force"
        ]
      },
      "N": {
        "title": "Normal force (N)",
        "description": "The force pressing the surfaces together, square to them.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "force",
        "minimum": 1e-12,
        "maximum": 1000000000000000
      },
      "m": {
        "title": "Mass (m)",
        "description": "The mass of the object resting on the surface.",
        "type": [
          "number",
          "string"
        ],
        "x-quantity": "mass",
        "minimum": 0.000001,
        "maximum": 1000000000000
      },
      "th": {
        "title": "Slope angle (θ, degrees)",
        "description": "The angle of the slope above level, from 0° to 90°.",
        "type": "number",
        "minimum": 0,
        "maximum": 90
      },
      "g": {
        "title": "Gravity (g, m/s²)",
        "description": "The acceleration of gravity: 9.80665 m/s² is standard on Earth.",
        "type": "number",
        "minimum": 0.001,
        "maximum": 1000
      }
    }
  },
  "outputs": {
    "result": {
      "label": "Answer",
      "description": "The friction force (N and lbf) or the coefficient, to 6 significant figures.",
      "format": "text"
    },
    "friction": {
      "label": "Friction force (N)",
      "description": "f = μ × N, in newtons: the kinetic force, or the largest static force.",
      "format": "number"
    },
    "lbf": {
      "label": "Friction force (lbf)",
      "description": "f in pounds-force.",
      "format": "number"
    },
    "mu": {
      "label": "Coefficient of friction (μ)",
      "description": "μ = f ÷ N, a pure number.",
      "format": "number"
    },
    "normal": {
      "label": "Normal force (N)",
      "description": "N, in newtons: m × g on level ground, m × g × cos θ on a slope.",
      "format": "number"
    },
    "angle": {
      "label": "Slope where it starts to slide (°)",
      "description": "arctan μ: with static friction, the steepest slope that holds; with kinetic, the slope of a steady slide.",
      "format": "number"
    },
    "along": {
      "label": "Weight along the slope (N)",
      "description": "m × g × sin θ: the pull of gravity down the slope.",
      "format": "number"
    },
    "slides": {
      "label": "Does it slide?",
      "description": "On a slope with static friction: whether μ_s N can hold the weight along the slope.",
      "format": "text"
    },
    "accel": {
      "label": "Acceleration down the slope (m/s²)",
      "description": "a = g × (sin θ − μ_k cos θ) for an object sliding down; negative means it slows.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "find": "force",
      "kind": "kinetic",
      "surf": "woodWood",
      "mu": 0.5,
      "f": "45 N",
      "by": "mass",
      "N": "196 N",
      "m": "20 kg",
      "th": 25,
      "g": 9.80665
    },
    "outputs": {
      "result": "f = 58.8399 N (13.2277 lbf)",
      "friction": 58.8399,
      "lbf": 13.227735731092654,
      "mu": 0.3,
      "normal": 196.133,
      "angle": 16.69924423399362
    },
    "text": "f = 58.8399 N (13.2277 lbf), with a normal force of 196.133 N."
  },
  "examples": [
    {
      "given": {
        "find": "force",
        "kind": "static",
        "surf": "custom",
        "mu": 0.7,
        "by": "mass",
        "m": 20,
        "g": 9.8
      },
      "expect": {
        "N": 196,
        "f": 137.2,
        "result": "f = 137.2 N (30.8438 lbf)"
      },
      "source": "OpenStax, University Physics Volume 1, §6.2 Friction (Equations 6.1 and 6.2; Table 6.1; Example 6.10: a 20.0 kg crate with μ_s = 0.700 and μ_k = 0.600 has N = 196 N, 137 N of static and 118 N of kinetic friction; Example 6.11: 45.0 N on a 62 kg skier on a 25° slope is μ_k = 0.082; Example 6.14: a = g(sin θ − μ_k cos θ) = 0.29 m/s² at 13° and μ_k = 0.20), https://openstax.org/books/university-physics-volume-1/pages/6-2-friction (retrieved 2026-10-19); hand calculation in content.mdx: N = 20 × 9.8 = 196 N, f_s ≤ 0.7 × 196 = 137.2 N"
    },
    {
      "given": {
        "find": "force",
        "kind": "kinetic",
        "surf": "custom",
        "mu": 0.6,
        "by": "force",
        "N": 196
      },
      "expect": {
        "f": 117.6
      },
      "source": "OpenStax, University Physics Volume 1, §6.2 Friction (Equations 6.1 and 6.2; Table 6.1; Example 6.10: a 20.0 kg crate with μ_s = 0.700 and μ_k = 0.600 has N = 196 N, 137 N of static and 118 N of kinetic friction; Example 6.11: 45.0 N on a 62 kg skier on a 25° slope is μ_k = 0.082; Example 6.14: a = g(sin θ − μ_k cos θ) = 0.29 m/s² at 13° and μ_k = 0.20), https://openstax.org/books/university-physics-volume-1/pages/6-2-friction (retrieved 2026-10-19); hand calculation in content.mdx: f_k = 0.6 × 196 = 117.6 N"
    },
    {
      "given": {
        "find": "coefficient",
        "kind": "kinetic",
        "f": 45,
        "by": "incline",
        "m": 62,
        "th": 25,
        "g": 9.8
      },
      "expect": {
        "mu": 0.08171824613777506,
        "result": "μ = 0.0817182"
      },
      "source": "OpenStax, University Physics Volume 1, §6.2 Friction (Equations 6.1 and 6.2; Table 6.1; Example 6.10: a 20.0 kg crate with μ_s = 0.700 and μ_k = 0.600 has N = 196 N, 137 N of static and 118 N of kinetic friction; Example 6.11: 45.0 N on a 62 kg skier on a 25° slope is μ_k = 0.082; Example 6.14: a = g(sin θ − μ_k cos θ) = 0.29 m/s² at 13° and μ_k = 0.20), https://openstax.org/books/university-physics-volume-1/pages/6-2-friction (retrieved 2026-10-19); hand calculation in content.mdx: μ_k = 45 ÷ (62 × 9.8 × cos 25°) = 0.0817 (OpenStax: 0.082)"
    },
    {
      "given": {
        "find": "force",
        "kind": "kinetic",
        "surf": "custom",
        "mu": 0.2,
        "by": "incline",
        "m": 60,
        "th": 13,
        "g": 9.8
      },
      "expect": {
        "accel": 0.2947550055908159
      },
      "source": "OpenStax, University Physics Volume 1, §6.2 Friction (Equations 6.1 and 6.2; Table 6.1; Example 6.10: a 20.0 kg crate with μ_s = 0.700 and μ_k = 0.600 has N = 196 N, 137 N of static and 118 N of kinetic friction; Example 6.11: 45.0 N on a 62 kg skier on a 25° slope is μ_k = 0.082; Example 6.14: a = g(sin θ − μ_k cos θ) = 0.29 m/s² at 13° and μ_k = 0.20), https://openstax.org/books/university-physics-volume-1/pages/6-2-friction (retrieved 2026-10-19); hand calculation in content.mdx: a = 9.8 × (sin 13° − 0.2 × cos 13°) = 0.29 m/s²"
    },
    {
      "given": {
        "find": "force",
        "kind": "static",
        "surf": "woodWood",
        "by": "incline",
        "m": 10,
        "th": 30,
        "g": 9.80665
      },
      "expect": {
        "th": 26.56505117707799,
        "slides": "Yes: it slides",
        "along": 49.03325
      },
      "source": "OpenStax, University Physics Volume 1, §6.2 Friction (Equations 6.1 and 6.2; Table 6.1; Example 6.10: a 20.0 kg crate with μ_s = 0.700 and μ_k = 0.600 has N = 196 N, 137 N of static and 118 N of kinetic friction; Example 6.11: 45.0 N on a 62 kg skier on a 25° slope is μ_k = 0.082; Example 6.14: a = g(sin θ − μ_k cos θ) = 0.29 m/s² at 13° and μ_k = 0.20), https://openstax.org/books/university-physics-volume-1/pages/6-2-friction (retrieved 2026-10-19), Table 6.1 (wood on wood: μ_s = 0.5); hand calculation in content.mdx: tan 30° = 0.577 > 0.5, so it slides; m g sin 30° = 49.03325 N"
    }
  ],
  "sources": [
    "OpenStax, University Physics Volume 1, §6.2 Friction: Equation 6.1 (f_s ≤ μ_s N), Equation 6.2 (f_k = μ_k N); Table 6.1 (approximate coefficients of static and kinetic friction); Example 6.10 (a 20.0 kg crate, N = 196 N, f_s up to 137 N, f_k = 118 N); Example 6.11 (45.0 N on a 62 kg skier on a 25° slope gives μ_k = 0.082); μ_k = tan θ at a steady slide; Example 6.14 (a_x = g(sin θ − μ_k cos θ) = 0.29 m/s²), retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/6-2-friction",
    "OpenStax, University Physics Volume 1, §5.6 Common Forces: the normal force N = mg on a level surface and N = mg cos θ on a slope; the weight along the slope mg sin θ, retrieved 2026-10-19. https://openstax.org/books/university-physics-volume-1/pages/5-6-common-forces",
    "NIST, CODATA 2022: standard acceleration of gravity g_n = 9.80665 m/s² (exact), retrieved 2026-10-19. https://physics.nist.gov/cgi-bin/cuu/Value?gn",
    "NIST Special Publication 811 (2008), Appendix B.8: 1 lb = 0.45359237 kg, 1 oz = 0.028349523125 kg, 1 lbf = 4.4482216152605 N, 1 kgf = 9.80665 N, retrieved 2026-10-19. https://www.nist.gov/pml/special-publication-811"
  ],
  "related": [
    "normal-force",
    "force",
    "centripetal-force",
    "tension"
  ],
  "changelog": []
}